Explanation

(a)
$\left| \begin{matrix}
\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} & \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} & \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{c}} \\
\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} & \overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} & \overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}} \\
\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} & \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} & \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}
\end{matrix} \right|$
$= \left| \begin{matrix}
a_{1}^{2} + a_{2}^{2} + a_{3}^{2} & a_{1}b_{1}{+ a}_{2}b_{2} + a_{3}b_{3} & a_{1}c_{1} + a_{2}c_{2} + a_{3}c_{3} \\
a_{1}b_{1}{+ a}_{2}b_{2} + a_{3}b_{3} & b_{1}^{2} + b_{2}^{2} + b_{3}^{2} & b_{1}c_{1} + b_{2}c_{2} + b_{3}c_{3} \\
a_{1}c_{1} + a_{2}c_{2} + a_{3}c_{3} & b_{1}c_{1} + b_{2}c_{2} + b_{3}c_{3} & c_{1}^{2} + c_{2}^{2} + c_{3}^{2}
\end{matrix} \right|$
$= \left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right|\left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right|$
$= \left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack^{2}$